competition between mixing and segregation in bimetallic ag n rb n clusters ( n = 2–10) 1,2

Competition between mixing and segregation inbimetallic AgnRbn clusters (n = 2–10)1,2

Rene Fournier, Shaima Zamiruddin, and Min Zhang

Abstract: We found the minimum-energy structures of AgnRbn (n = 2–10) clusters by a combination of density functionaltheory (DFT) and taboo search global optimization. The global minimum geometry is mixed for n £ 4 and segregated, witha core-shell arrangement, for n > 4. There is a change in the nature of the bonding, from ionic to metallic, between n = 4and n = 5. Although metallic bonding dominates at n > 4, large atomic charges (in the order of ±0.5) persist. These atomiccharges (negative on the interior Ag atoms, positive on the surface Rb atoms) make AgnRbn clusters analogous to Zintl com-pounds and could prevent them from coalescing. This makes them intriguing potential building blocks for cluster-assembledmaterials. Ag4Rb4 is relatively stable compared with other AgnRbn clusters; it has a nearly cubic shape, a large HOMO–LUMO gap (2 eV), and a highly ionic character with atomic charges equal to roughly ±1 au.

Key words: bimetallic clusters, nanostructured materials, global energy minima, segregation, Zintl compounds.

Resume : Nous avons determine les structures d’energie minimum d’agregats AgnRbn (n = 2–10) par une methode qui com-bine la theorie de la fonctionelle de la densite avec une recherche de minima par optimisation globale ( recherche tabou ).Les minima globaux que nous obtenons sont des structures mixtes d’alliages pour n £ 4, et des structures segregationnee detopologie concentrique Agn@Rbn, pour n > 4. La nature des liens evolue en fonction de n, avec de fortes interactions ioni-ques pour n £ 4 et des liens de type metallique pour n >4. Cependant, des charge atomiques appreciables subsistent a n > 4(charges negatives sur les atomes de Ag qui sont a l’interieur, et charges positives sur les atomes de Rb en surface). La sepa-ration radiale des charges rend les agregats de AgnRbn analogues aux composes Zintl, et les rend probablement stables vis-a-vis la coalescence. Cette propriete ferait des agregats de AgnRbn des unites de base interessantes pour d’eventuels mate-riaux. Ag4Rb4 est relativement stable compare aux autres agregats de AgnRbn et possede une structure cubique ainsi qu’uneforte separation des niveaux d’energie des orbitales frontieres (2 eV) et un fort caractere ionique avec des charges atomiquesde l’ordre de ±1 au.

Mots-cles : agregats bimetalliques, materiaux a l’echelle du nanometre, minima globaux d’energie, segregation, composesZintl.

Introduction

The structure of small metal clusters, and particularly‘‘bimetallic’’ clusters, are not well-known. The phase dia-grams of bulk alloys do not give reliable predictions aboutpossible mixing in the corresponding small bimetallic clus-ters.3 For alloying, as for many other things, clusters areoften very different from the bulk. Many simple questionsabout bimetallics AnBp are still open. Generally, one wouldlike to know if A and B segregate, or if they mix to form ananoalloy. Experimentally, the structures of bimetallic clus-ters AnBp often fall into one of these three categories: mixed,core-shell, or onion-layered.4 There is also a fourth category,‘‘left-right segregated’’, that often occurs in the bulk limitbecause it minimizes the A–B interfacial area. A recent com-

putational study showed that Pd–Pt clusters described by aGupta potential adopt different types of segregated structuresdepending on the value of A–B interaction parameters usedin the potential.5 One could also distinguish between variousmixed structures, some of them ordered, others, random. Infact, the details of A–B ordering can be intricate even insimple theoretical models of coinage (Cu, Ag, Au) bimetallicclusters.6

We chose to study the series of clusters AgnRbn (n = 2–10)as a model to test computational methods and tackle generalquestions about AB order in bimetallic clusters because(a) it is simple (Ag and Rb atoms have only s valence elec-trons), (b) the big difference in absolute electronegativity c =(I + A)/2 (I is the ionization energy, A is the electron affinity,c = 4.44 eV for Ag, and c = 2.33 eV for Rb) favors mixingbut, (c) the big difference in cohesive energies (2.95 eV forAg, 0.85 eV for Rb) and atomic radii (1.45 A for Ag, 2.42 Afor Rb) favor core-shell segregation. We expect (c) to domi-nate and control the structure at large n. But small clusters,where there are few or no interior atoms, could be very dif-ferent, and the size evolution of structure is non-trivial. Thenature of the bonding in these small clusters is also unclear apriori. Ag and Rb both have a very low ionization energy,and many aspects of Agn

7 and Rbn8 clusters are reasonably

Received 18 December 2008. Accepted 19 March 2009.Published on the NRC Research Press Web site atcanjchem.nrc.ca on 23 June 2009.

R. Fournier1 and S. Zamiruddin. Department of Chemistry,York University, Toronto, ON M3J 1P3, Canada.M. Zhang. Department of Physics, York University, Toronto,ON M3J 1P3, Canada.

1Corresponding author (e-mail: [email protected]).

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Can. J. Chem. 87: 1013–1021 (2009) doi:10.1139/V09-065 Published by NRC Research Press

well-described in the jellium model.9 So, in a way, Ag andRb are ideal metals. But in AgnRbn, ionic interactions couldbe important, and maybe small clusters of AgnRbn are noteven metallic, after all. Clusters like AgnRbn are interestingfor another reason. Their presumed core-shell structure, com-bined to large surface charges, should prevent them from co-alescing. This is important for making cluster-assembedmaterials. For instance, size-selected metastable Agn@Rbn orAgn@Nan clusters (the @ symbol denotes a core-shell struc-ture with the Ag atoms inside) could be relatively stable in-termediates on the way to making uniform size silverclusters embedded in ionic solids (e.g., RbCl, NaCl). In fact,one can view Agn@Rbn as a metal cluster analog of Zintlcompounds, with a true metal (Ag) in the interior instead ofa semimetal (Ge, Sn, Pb, Sb, and so forth) like in the knownZintl compounds,10 so there is a slight possibility that it couldbe stable in itself.

MethodsWe performed global optimization by the taboo search in

descriptor space (TSDS) method.11 All energies were calcu-lated by density functional theory (DFT) with the Perdew–Burke–Ernzerhof (PBE)12 generalized gradient approxima-tion to exchange-correlation and a LANL2 double-zeta(LANL2DZ) basis set as implemented in the Gaussian 03software package.13 The number of possibles isomers (localminima) for elemental clusters Xn is quite large; for a 13-atom cluster, it is in the order of 1000, and the number of iso-mers roughly triples for every additional atom.14 However,the number of isomers for a bimetallic cluster AnBn is muchgreater than for an elemental cluster X2n. Ignoring symmetry,there are (2n)!/(n!)2 ways to substitute n atoms of X by atomsof A (and the rest by atoms of B). The (2n)!/(n!)2 isomers ofAnBn that result from these different ways to occupy siteswith atoms of type A or B are often called ‘‘homotops’’.15

For Ag10Rb10, then, an order of magnitude estimate for thenumber of isomers is 1011. Fortunately, the optimizationproblem is not as difficult as it seems because the vast major-ity of isomers are very high in energy, and the low-energyisomers fall into one or a few categories that share commongeometric characteristics. The TSDS algorithm exploitsthese resemblences among low-energy structures to quicklyidentify regions of configuration space where the search islikely to succeed, and it finds GMs, or good approximationsto them, with very few energy evaluations even in difficultcases.16,17 The TSDS runs for this study consisted of (12n2 +30) iterations, where 2n is the number of atoms. At everyiteration, TSDS produces a structure of AgnRbn with prede-fined nearest-neighbour (NN) distances. A few preliminaryruns of local optimization on arbitrary trial geometries ofsmall AgnRbn clusters show what distances are appropriate,and they were set at 2.82 A for Ag–Ag, 4.70 A for Rb–Rb,and 3.65 A for Ag–Rb. For comparison, the bulk experimen-tal interatomic distances are 2.89 A for Ag–Ag and 4.84 Afor Rb–Rb. Other control parameters in TSDS ensure thatthe majority of the (12n2 + 30) structures generated in thisway are topologically distinct, but a fraction of them are‘‘distorted copies’’ of previous structures. These (12n2 + 30)geometries are not local minima of the potential surface, butare near local minima. For this reason, our optimization algo-rithm can be viewed as having an additional approximation

compared to basin hopping (BH) methods,18 but having theadvantage of being much more economical (by roughly a fac-tor of 50) compared to BH in terms of the number of energyevaluations required. The last point is, of course, crucial fordoing global optimization directly on a first-principles DFTenergy surface. After the TSDS iterations are complete, the(12n2 + 30) structures are ordered according to their energyand their similarity to higher ranked structures, i.e., similarstructures get pushed down the ranks. The 10 to 20 top inthis order constitute the lowest energy-distinct structures: wetake them as initial geometries for local optimization withGaussian 03 PBE/LANL2DZ.

To analyze trends in the final optimized structures, we usea number of geometric descriptors. Note that these are, gen-erally speaking, different from the descriptors used duringTSDS optimization. The three basic descriptors that we usefor characterizing AB order are the number of A–B NNpairs m, and two variables, c and ‘, that quantify core-shellsegregation (CSS) and left–right segregation (LRS), respec-tively. In the following equations, a and b are indices thatlabel atoms of type A and B, respectively, dab is the distancebetween a A–B pair of atoms, RA and RB are the atomicradii for elements A and B, RAB = RA + RB, P

!a and P

!b

are position vectors of the nuclei, and C!

is the center ofmass of the cluster calculated with fictitious masses of onefor all atoms. We use a simple cut-off function of intera-tomic distance f(dab) to decide if two atoms are neighbours,and add up to get the number of AB neighbour pairs.

½1�

f ðdabÞ ¼ 1 if dab < 1:1RAB

¼ ½1:3RAB � dab�=0:2RAB if 1:1 � dab=RAB � 1:3

¼ 0 if 1:3RAB < dab

m¼ 1=2Xn

a¼1

Xn

b¼1

f ðdabÞ

The two segregation variables are simple relations be-tween centers of mass of different types of atoms.

½2� ‘ ¼ 1

n

Xn

a¼1

P!

a �Xn

b¼1

P!

b

��

½3� c ¼ 1

n

Xn

a¼1

jP!a � C!j �

Xn

b¼1

jP!b � C!j

" #

We normally do not use m, ‘, or c directly. Instead, wecalculate m, ‘, and c for all the homotops of a given struc-ture, or a large sample of those homotops, and estimate theaverage ‘‘Avg’’ and standard deviation ‘‘SD’’ of m, ‘, and cover the sample set of homotops. We sample homotops bydoing a long sequence of random A–B position inter-changes, calculating m, ‘, and c each time. We denote thevalues of the variables for the structure itself (parent homo-top) m0, ‘0, and c0, and define relative descriptors like this:

½4� M ¼ m0 � AvgðmÞSDðmÞ

½5� L ¼ ‘0 � Avgð‘ÞSDð‘Þ

1014 Can. J. Chem. Vol. 87, 2009

Published by NRC Research Press

½6� C ¼ c0 � AvgðcÞSDðcÞ

So, for instance, an M value of +2.00 means that thisstructure has a number of A–B nearest neighbours that islarger than the average value for all of its homotops by twostandard deviations; a value of zero for ‘ means that thestructure has a degree of left–right segregation that is aver-age relative to the set of all homotops to which it belongs,and so forth. Another variable that we use to quantify core-shell separation is the number of interior atoms Ni defined asfollows: First, we calculate the asphericity zj around eachatom j by

½7� zj ¼ðIa � IbÞ2 þ ðIb � IcÞ2 þ ðIc � IaÞ2

I2a þ I2

b þ I2c

where Ia, Ib, and Ic are the moments of inertia of the neigh-bours of atom j. We assign, to atom j, a number nj = exp(–(zj/z0)2), which tends to be close to one for interior atomsand close to zero for surface atoms, and get an effectivenumber of interior atoms, Ni, by adding up the nj’s.

½8� Ni ¼X

j

nj ¼X

j

exp ð�ðzj=z0Þ2Þ

With these definitions and z0 = 0.0601, we almost alwaysget nj > 0.9 for atoms that we would have classified as ‘‘in-terior atoms’’ upon visual inspection and nj < 10–5 for typicalsurface atoms. Another descriptor of geometry that we use isF90, the number of 908 angles per atom. It is defined by

½9� F90 ¼1

2n

Xj

Xk>‘

cosðqkjl � 90�Þ100

where k and ‘ index the neighbours of atom j. The exponent100 in that formula is arbitrary, but it produces an intui-tively correct count of the number of right angles.

Results and discussionA taboo search explicitly encourages diverse solutions;

therefore, it not only has a good chance of finding the globalminimum (GM), but it is also likely to find many of the low-est-energy isomers. This is illustrated in Fig. 1, which showsthe nine isomers found for Ag3Rb3 (another structure foundby TSDS turned out to be a repeat of the 0.080 eV isomer).We should say a few words to put our results in perspective.We estimate that DFT-PBE gives errors of roughly 0.1 eV to0.2 eV on relative cluster isomer energies for elements withlow cohesive energy like Ag and Rb, and we believe that itis unlikely that TSDS would miss the GM but it is morelikely that it would miss some (and maybe many) of thelow-energy isomers. That said, there is clearly a lot of diver-sity among the lowest-energy structures of Ag3Rb3 some flatand some tridimensional, some well-mixed (e.g., 0.000 eV)and some segregated (e.g., 0.324 eV). The same is true forAg5Rb5 (four very different isomers found within 0.1 eV,see Fig. 2) and Ag6Rb6 (three very different isomers foundwithin 0.1 eV). The Ag9Rb9 and Ag10Rb10 clusters also havemany isomers within 0.1 eV of the GM, but those isomers aresimilar, all of them have a silver core and an incomplete ru-bidium shell. In Fig. 3, we show the distribution of isomerenergies as a function of n, and the cohesive energies of GMas a function of N–1/3 (N = 2n). The number of isomers withina small energy range of the GM varies a lot. We did not findany isomer within 0.2 eV of the GM at n = 2, 4, and 8 butfound five for n = 3, six for n = 5, four for n = 6, one for n =7, four for n = 9, and six for n = 10 (some of them are indis-tinguishable in Fig. 3a).

Notwithstanding the uncertainty about the identity of theGM, it is convenient to focus on the structures and proper-ties of the putative GM to see how they evolve with size(Figs. 3 and 4). Cohesive energies E(N) of the GM(Fig. 3b) follow a typical trend for metal clusters; they arewell-fitted by the function E(N) = Ebulk + bN–1/3, with a

Fig. 1. Isomers of Ag3Rb3 and their relative energies (eV).

Fournier et al. 1015


correlation coefficient of –0.9918. The intercept Ebulk is1.75 eV/atom, which is 8% smaller than the average of ex-perimental cohesive energies for Ag (2.95) and Rb (0.85).The negative of the slope (1.26) is related to surface energy:it is much smaller than the negative of the slope (3.48) thatwe got by fitting the Agn cohesive energies obtained yearsago in another series of DFT calculations.7 This is not sur-prising considering that the surfaces of AgnRbn clusters areenriched in Rb atoms. Generally, for elemental solids andclusters, surface energy is roughly proportional to (Ec/R2),

where Ec is the cohesive energy and R is the bulk intera-tomic distance. So, we can get a rough estimate of whatthe slope would be for pure Rb clusters by taking 3.48 �(0.85/2.95) � (2.89/4.837)2, which is 0.36. So, the slope of1.26 for the mixed Ag–Rb clusters must surely be very dif-ferent from the pure Rbn case also. Looking at Ni for n = 5to 10, we see that it is roughly n/4 on average, i.e., a quar-ter of Ag atoms are ‘‘interior atoms’’ in that size interval.The weighted average ð3=4 � 3:48þ 0:36Þ=1:75 ¼ 1:70 is rea-sonably close to 1.26, so the slope of the fit line in Fig. 3b

Fig. 2. Isomers of AgnRbn (n > 3) within 0.10 eV of the GM and their relative energies.

1016 Can. J. Chem. Vol. 87, 2009


can be understood as a sort of weighted average of the sur-face energies of Ag and Rb. At very large n, the surfaceshould consist entirely of Rb atoms, and the slope of astraight line fit would have to be close to –0.36, not –1.26.So, clearly, a linear function of N–1/3 cannot fit accuratelythe cohesive energies of AgnRbn over the entire range of n.

The n = 2–4 GM structures are very well-mixed (Figs. 4a–4c). A natural bond orbital (NBO) analysis reveals atomiccharges of nearly ±0.75 in all three cases and shows thatbonding is predominantly ionic. The structures in biggerclusters show a strong tendency for Ag atoms to be near thecenter, and by n = 8, the silver core is already fairly compactand tridimensional. The NBO analysis for n > 4 shows thatcharges are smaller (~ ± 0.50) and that a larger fraction ofthe valence electrons are delocalized, both of which indicatea transition from ionic to metallic bonding at around n = 5.

We also calculated atomic charges using the self-consis-tent charge equalization (SC QEq) method,6,19 which is animproved version of QEq,20 to gain more insight. In SCQEq, an electronegativity parameter c is assigned to each el-ement (Ag, Rb), along with parameters for the first, second,and third derivatives of c with respect to atomic charge.These derivatives also depend on an atom’s position, throughits screened Coulomb interactions to other atoms, so, for in-stance, the Ag atoms in a given cluster do not necessarily allend up with the same charge. Atomic charges, and charge-dependent atomic electronegativities, are calculated itera-tively until they satisfy the principle of electronegativityequalization.21,22 An approximate electrostatic energy is thencalculated by summing pairwise-screened Coulomb interac-

tions. Details of the method can be found in the literature.19

For the GM of AgnRbn (n = 2, 3, and 4), the average SCQEq charges on Rb atoms are 0.76, 0.91, and 1.12, respec-tively. For the GM at n = 5–9, the average SC QEq chargeson Rb atoms vary between 0.40 and 0.53 and have no simplerelation to n, and for n = 10, it is 0.27. These SC QEqcharges are not very different from the NBO charges. TheSC QEq electrostatic energy per atom is strongly correlatedto the average magnitude of SC QEq charges. So, the differ-ent geometric AB ordering of the GMs as a function of n,along with a simple electrostatic model that ignores quantummechanical effects (SC QEq), are sufficient to explain somequalitative differences in bonding seen in a more rigorous(NBO) analysis. Other interesting things emerge from theSC QEq analysis. First, at n > 4, the average magnitudes ofSC QEq charges in the GM are neither small or big com-pared to those in other structures found in our TSDS search.But for n £ 4, the average magnitudes of SC QEq charges inthe GMs are much bigger than in the other structures, whichindicates ionic bonding in the GMs, and they increase as afunction of size (from n = 2 to n = 4), whereas other struc-tures show no clear trend with size. Second, the SC QEqelectrostatic energies of the different isomers for a givencluster size vary over a big range: 1.1 eV (n = 2), 2.1 eV(n = 3), 3.5 eV (n = 4), 1.8 eV (n = 5), and for n > 5, atypical range is n/2 eV. At n > 4, the GMs have SC QEqelectrostatic energies that are typically 2–3 eV less negative(less stable) than for the structure with most negative SCQEq electrostatic energy for that size. Third, there is a bigdrop in the SC QEq average Rb charge between n = 9(0.47) and n = 10 (0.27) and, linked to that, a big drop inthe mixing parameter M (Fig. 5c). One could say thatAgnRbn clusters (n £ 10) are an example of a frustrated sys-

Fig. 3. Relative isomer energies, and cohesive energies of the GMs. Fig. 4. Global minima.



tem. Their lowest-energy structures do not necessarily mini-mize electrostatic energy, or surface energy (i.e., do notmaximize the number of nearest neighbours). Instead, theGM structures are normally the best compromise betweenthese two things, and evolve with size from mixed (n £ 4)to ‘‘mostly core-shell segregated’’ (n = 10). We wanted tosee how much difference there is between the GM of AgnRbn(n > 4) and ideal CSS structures. To do that, we calculatedthe mean coordination ‘‘x’’ in hypothetical Agn clusters thatconsist of the n Ag atoms at their positions in the GM ofAgnRbn, x(Ag;AgnRbn); and for comparison, we calculatedthe mean coordinations of Ag atoms in the GM of Agn clus-ters that we had previously obtained,7 x(Ag;Agn). In eithercase, x is the mean number of Ag atom neighbours around aAg atom. An ideal AgnRbn CSS structure would have a cen-tral compact Agn subunit with a structure comparable to, androughly as compact as, that of a pure Agn cluster. What wefind, going in order from n = 2 to n = 10, is x(Ag;AgnRbn) =0.0, 0.0, 0.0, 2.4, 2.3, 3.4, 3.5, 3.3, and 4.6. These values aremuch smaller than the corresponding x(Ag;Agn); going againin order from n = 2 to n = 10, we have x(Ag;Agn) = 1.0, 2.0,2.5, 2.8, 3.0, 4.6, 4.5, 4.7, and 5.2. We see that the Agn sub-units in AgnRbn are a lot less compact than Agn clusters, ex-cept maybe for n = 5 and n = 10. This again shows that theGM of AgnRbn (n > 4) represent a compromise betweenminimizing surface energy and maximizing ionic energy(mixing).

The size evolution of the HOMO–LUMO gap and somegeometric descriptors of the GMs are shown in Fig. 5. Thepeak in HOMO–LUMO gap at n = 4 is remarkable. Two otherthings indicate special stability for Ag4Rb4: the E(n = 4) point

is well above the fit line of cohesive energies (Fig. 3b), andthe second most stable isomer found for Ag4Rb4 is muchhigher (0.3 eV) than the GM (Fig. 3a). It is tempting to as-cribe that to the eight-electron count, which is a closed shellin the jellium model. However, the difference in bulk elec-tron densities of Ag and Rb, 0.0087 – 0.0017 = 0.0070 au,is very close to the critical value (0.0080 au) beyond whichone normally sees inversion of the 2s and 1d jellium levelsand magic numbers of 2, 10 (instead of 8), and 20.3, 23 Also,the large atomic charges (NBO and SC QEq) in Ag4Rb4 in-dicate that ionic bonding is more important than delocalizedmetallic bonding. The SC QEq electrostatic energy, which isadmittedly very approximate, amounts to 60% of the KS-DFT-PBE atomization energy of Ag4Rb4. Aside from theanomaly at n = 4, and maybe n = 10, HOMO–LUMO gapsdecrease smoothly from n = 2 to n = 10, suggesting a grad-ual transition from ionic to metallic bonding.

There is a rationale for why the transition from mixed toCSS structure should occur near n = 5. Ionic bonding favorsmixed structures at any size. On the other hand, the differ-ence in surface energies between Ag and Rb, which isroughly EsðAgÞ � EsðRbÞ � 0:4� ð2:95 � 0:85Þ ¼ 2:1 eVper surface atom, favors CSS structures, i.e., it disfavorsmixing, but only provided that the number of interior atoms(Ni) is not zero of course. Generally speaking, for metalclusters, one expects that Ni becomes nonzero for clusterswith more than roughly 12 atoms because the 13-atom ico-sahedron is the smallest cluster in the polytetrahedral growthsequence with one interior atom. For structures other thanpolytetrahedral, Ni can become nonzero at slightly smalleror slightly bigger size. In mixed AnBn clusters where the el-

Fig. 5. Size evolution of structure and properties in the GM of AgnRbn.

1018 Can. J. Chem. Vol. 87, 2009


ement with lowest cohesive energy has a bigger atomic ra-dius, like AgnRbn, the smallest cluster size with one interioratom can occur at a smaller size, 10 atoms (n = 5) in ourcase (see Fig. 5b). As soon as clusters are big enough to ac-commodate one interior atom, the maximally mixed struc-ture is no longer automatically favored, and the GM aregenerally ‘‘structures of compromise’’. Using a crude model(see Appendix 1), we estimate that the surface energy of aCSS structure is lower than that of a mixed (ionic) structureby an amount DUs that is roughly (2n – 8) � 0.43 eV inAgnRbn clusters (n > 4). On the other hand, as we said ear-lier, the range in calculated SC QEq electrostatic energiesamong different isomers of AgnRbn is roughly 0.5n eV. So,these two contributions to energy differences between mixedand segregated structures have comparable magnitudes, andthey both scale roughly as n, which gives a rationale forwhy, starting with n = 5, the structures of GM are neitherof the mixed nor the ideal (compact) core-shell type.

There remains a strong ionic character (charges of ±0.5)even in the largest clusters. This has an effect on the pre-ferred geometries of AgnRbn clusters. In particular, the num-ber of 908 bond angles (F90) per atom is large compared withother metal clusters: F90 varies between 1 and 4 for AgnRbn(n > 3), and it is 2.1 for n = 6 (12-atom cluster) and 2.9 for n= 7 (14-atom cluster). The average of 2.1 and 2.9, 2.5, can becompared to the F90 value we found for the calculated GM of26 different 13-atom metal clusters.17 Of those 26 cases, onlysix (Tc13, Ru13, Rh13, Re13, Os13, Ir13) had F90 > 2.5. Thelarge F90 values in those six cases were explained by the par-ticipation of d orbitals in bonding; but this explanation obvi-ously cannot apply to AgnRbn. The largest F90 wecalculated17 for 13-atom clusters of group 1, 2, 11, or 12 inthe periodic table was 1.5 (for Zn13), much smaller than 2.5.We think that the large F90 in AgnRbn is a consequence of thelarge ionic contribution to bonding and the effect this has onA–B ordering, since cubic packing (as in NaCl) would mini-mize the electrostatic energy. The tendency to adopt struc-tures with large F90 is not limited to the GM of AgnRbn;several isomers (some of them not shown here) have pris-matic or square arrangements of Rb atoms, e.g., Fig. 2g and2k. It is also interesting to track the number of interior atoms,Ni, as a function of n (see Fig. 5b), but we are unable to pre-dict accurately the size where Ni = n, i.e., the smallest clusterwith n interior Ag atoms covered by a ‘‘monolayer’’ of n Rbatoms (this would correspond to the closure of an atomicshell in the core-shell growth sequence and could give en-hanced stability). A simple geometric model leads us to pre-dict that, within the AgnRbn sequence, the Rb shell of theatoms becomes complete only at n = 30, roughly.

The mixing descriptor M and core-shell segregation pa-rameter C (Figs. 4j and 4k) both undergo a sudden changebetween n = 4 and n = 5. The value of M starts off verylarge (at n = 2) and goes down but remains above zero untiln = 10. As expected, the large electronegativity differencebetween Ag and Rb favors mixing. The drop of M at n =10 shows that the driving force for having Ag atoms inside(difference in surface energies of Ag and Rb) is bigger thanthe driving force for mixing (difference in electronegativitiesof Ag and Rb). There is no special significance to M becom-ing negative at precisely n = 10. It happens partly because,with increasing number of atoms, there are many more ways

to make highly mixed homotops, which makes the CSSstructures appear poorly mixed in a relative sense (seeeq. [4]). The C values get very large at n > 4 (+2.0 to +4.0)while L always stays negative; this shows that AgnRbn hasindeed a strong tendency to form core-shell structures, evenfor n values as small as 5.

The shape descriptor, h = (2Ib – Ia – Ic)/Ia, of the GMs ofAgnRbn is negative for 2n = 4, 6, 14, 16, and 18, positive for2n = 10 and 12, and very nearly zero for 2n = 8 and 20.These predictions of oblate, prolate, and spherical shapesagree with the ellipsoidal jellium model in every case exceptAg2Rb2. The peaks in the HOMO–LUMO gap at 2n = 8 and20 also agree with the jellium model, but this is partly coin-cidental. For instance, the structure of Ag4Rb4 is optimal forionic bonding (it is cubic), and also for accomodating aspherical closed-shell of eight electrons (Ia = Ib = Ic) as perthe jellium model, but bonding in Ag4Rb4 is predominantlyionic. However, the agreement of DFT and EJM clustershapes for all cases at n ‡ 5 is probably not entirely acciden-tal either. The EJM may be a crude model of electronicstructure, but it gives a qualitatively correct relation betweencluster shape and orbital symmetry and energies. These ef-fects of shape and orbital symmetry are implicitly includedin a more complete theory like DFT and may well give asmall energetic preference (maybe in the order of a fewtenths of eV) to structures that conform to the EJM.

In conclusion, the structures and nature of chemical bond-ing in AgnRbn clusters evolve from mixed AB structureshaving ionic bonding for n £ 4 to CSS structures where met-allic bonding dominates but where ionic bonding is still im-portant for n = 5 to 10. The Ag4Rb4 clusters are special fortheir relative stabilities and large HOMO–LUMO gaps. Thegeometric structures of AgnRbn (n = 5 to 10) have a Ag coreand an incomplete Rb shell, but the Ag core is not verycompact. These structures are the result of a compromise be-tween metallic and ionic contributions to bonding. Atomiccharges on the Rb outer-shell atoms in Ag10Rb10 are calcu-lated to be +0.3; this may be large enough to prevent coales-cence of larger Agn@Rbn clusters having a complete Rbouter shell.

AcknowledgementsRF is grateful to the European Science Foundation (ESF)

for supporting the 2008 Workshop on ComputationalNanoalloys where participants discussed many ideas that arerelevant to the work presented here, and to the NaturalSciences and Engineering Research Council of Canada(NSERC) for research funding.

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Appendix I. Model of surface energy in mixedand core-shell AnBn clusters

We want to estimate the energy difference, DUs, arisingfrom the different surface area, structure, and compositionat the surface of two qualitatively different types of AnBncluster structures (ionic and maximally mixed (IMM), andcore-shell (CS)). A model like SC QEq could be used to es-timate energy differences due to electrostatics, DUe. Thesign of DUs + DUe could then be used to make qualitativepredictions of cluster structure in AnBn mixed clusters.

We assume that IMM clusters have a packing similar tothe simple cubic crystal structure and have a cubic shape,whereas the CS clusters are spherical with atoms packed ina way similar to fcc. These assumptions clearly cannot beexact for small clusters, but they hopefully capture an essen-tial difference in the surface-to-volume ratio for the twotypes. So, the volume V and surface area A of the two struc-ture types are estimated to be

½A1�IMM V ¼ Nv

ffiffiffi2

pA ¼ 6V2=3 ¼ 7:560ðNvÞ2=3

� �

CSV ¼ Nv

A ¼ 4:836V2=3 ¼ 4:836ðNvÞ2=3� �

where N = 2n is the number of atoms and v is the averageof the bulk atomic volumes for elements A and B. The sur-face of an IMM cluster is assumed to have A–B composi-tion of 50%:50%, so the surface energy for IMM can beestimated as

½A2� Us ¼ ðAIMM=aÞEc � 0:4

where Ec is the average of the two cohesive energies((0.85 + 2.95)/2 = 1.90 eV for Ag–Rb) and a is the squareof the average of bulk A–A and B–B distances (14.9 A2 forAg–Rb); a is a rough estimate of surface area per atom in amixed ordered A–B surface. The factor 0.4 is empirical; afactor of 0.16 would be appropriate for surface atoms ofmacroscopic particles,24 but the cohesive energy of verysmall metal clusters (those with Ni = 0) is often close to60% of the bulk cohesive energy, which motivates thechoice of 0.4. The composition of the surface of a CS clus-ter depends on the number Ni of interior atoms. If Ni is largeenough, the surface of CS clusters have only atoms of type‘‘B’’ (the one with lowest surface energy), and in this parti-cular case, the surface energy of the CS cluster would be

½A3� Us ¼ ðACS=R2BÞEc;B � 0:4

Let us define fm = 7.56Ec/a and fc ¼ 4:836Ec;B=R2B. Then,

the surface energy for IMM is fm � 0.4 � (Nv)2/3, and thesurface energy of a very large CS cluster is fc � 0.4 �(Nv)2/3. For the Ag–Rb case, 0.4 � v2/3 = 4.78 A2. When aCS cluster is not too large, there are Ni < n interior atoms,and the surface composition is not 100% B. Instead, it iscloser to (N/2 – Ni) atoms of A and N/2 atoms of B, or alter-

1020 Can. J. Chem. Vol. 87, 2009


natively, (N – 2Ni) of AB, and Ni of B. So, we calculate afactor fx for intermediate size CS cluster like this:

½A4� fx ¼ ½ðN � 2NiÞfm þ Nifc�=ðN � NiÞ

and estimate the surface energy of the CS cluster as

½A5� Us ¼ fx � 0:4� ðNvÞ2=3¼ fx � 4:78� N2=3 for Ag�Rb

From our calculated Ni values (Fig. 5b), we can propose asimplified approximate relation between Ni and N, Ni &(N – 4)/2. Using this relation and the equations above, wecan estimate the contribution DUs to the energy differencebetween IMM- and CS-type structures that is specificallydue to surface structure and composition. We find DUs = 0,1.8, 3.4, and 4.9 eV for N = 8, 12, 16, and 20, respectively.



competition between mixing and segregation in bimetallic ag n rb n clusters ( n = 2–10) 1,2

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